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<div class="titlepage"><div><div><h3 class="title">
<a name="boost_numeric_odeint.concepts.symplectic_system"></a><a class="link" href="symplectic_system.html" title="Symplectic System">Symplectic
      System</a>
</h3></div></div></div>
<h5>
<a name="boost_numeric_odeint.concepts.symplectic_system.h0"></a>
        <span class="phrase"><a name="boost_numeric_odeint.concepts.symplectic_system.description"></a></span><a class="link" href="symplectic_system.html#boost_numeric_odeint.concepts.symplectic_system.description">Description</a>
      </h5>
<p>
        This concept describes how to define a symplectic system written with generalized
        coordinate <code class="computeroutput"><span class="identifier">q</span></code> and generalized
        momentum <code class="computeroutput"><span class="identifier">p</span></code>:
      </p>
<p>
        <span class="emphasis"><em>q'(t) = f(p) </em></span>
      </p>
<p>
        <span class="emphasis"><em>p'(t) = g(q) </em></span>
      </p>
<p>
        Such a situation is typically found for Hamiltonian systems with a separable
        Hamiltonian:
      </p>
<p>
        <span class="emphasis"><em>H(p,q) = H<sub>kin</sub>(p) + V(q) </em></span>
      </p>
<p>
        which gives the equations of motion:
      </p>
<p>
        <span class="emphasis"><em>q'(t) = dH<sub>kin</sub> / dp = f(p) </em></span>
      </p>
<p>
        <span class="emphasis"><em>p'(t) = dV / dq = g(q) </em></span>
      </p>
<p>
        The algorithmic implementation of this situation is described by a pair of
        callable objects for <span class="emphasis"><em>f</em></span> and <span class="emphasis"><em>g</em></span> with
        a specific parameter signature. Such a system should be implemented as a
        std::pair of functions or a functors. Symplectic systems are used in symplectic
        steppers like <code class="computeroutput"><span class="identifier">symplectic_rkn_sb3a_mclachlan</span></code>.
      </p>
<h5>
<a name="boost_numeric_odeint.concepts.symplectic_system.h1"></a>
        <span class="phrase"><a name="boost_numeric_odeint.concepts.symplectic_system.notation"></a></span><a class="link" href="symplectic_system.html#boost_numeric_odeint.concepts.symplectic_system.notation">Notation</a>
      </h5>
<div class="variablelist">
<p class="title"><b></b></p>
<dl class="variablelist">
<dt><span class="term"><code class="computeroutput"><span class="identifier">System</span></code></span></dt>
<dd><p>
              A type that is a model of SymplecticSystem
            </p></dd>
<dt><span class="term"><code class="computeroutput"><span class="identifier">Coor</span></code></span></dt>
<dd><p>
              The type of the coordinate <span class="emphasis"><em>q</em></span>
            </p></dd>
<dt><span class="term"><code class="computeroutput"><span class="identifier">Momentum</span></code></span></dt>
<dd><p>
              The type of the momentum <span class="emphasis"><em>p</em></span>
            </p></dd>
<dt><span class="term"><code class="computeroutput"><span class="identifier">CoorDeriv</span></code></span></dt>
<dd><p>
              The type of the derivative of coordinate <span class="emphasis"><em>q'</em></span>
            </p></dd>
<dt><span class="term"><code class="computeroutput"><span class="identifier">MomentumDeriv</span></code></span></dt>
<dd><p>
              The type of the derivative of momentum <span class="emphasis"><em>p'</em></span>
            </p></dd>
<dt><span class="term"><code class="computeroutput"><span class="identifier">sys</span></code></span></dt>
<dd><p>
              An object of the type <code class="computeroutput"><span class="identifier">System</span></code>
            </p></dd>
<dt><span class="term"><code class="computeroutput"><span class="identifier">q</span></code></span></dt>
<dd><p>
              Object of type Coor
            </p></dd>
<dt><span class="term"><code class="computeroutput"><span class="identifier">p</span></code></span></dt>
<dd><p>
              Object of type Momentum
            </p></dd>
<dt><span class="term"><code class="computeroutput"><span class="identifier">dqdt</span></code></span></dt>
<dd><p>
              Object of type CoorDeriv
            </p></dd>
<dt><span class="term"><code class="computeroutput"><span class="identifier">dpdt</span></code></span></dt>
<dd><p>
              Object of type MomentumDeriv
            </p></dd>
</dl>
</div>
<h5>
<a name="boost_numeric_odeint.concepts.symplectic_system.h2"></a>
        <span class="phrase"><a name="boost_numeric_odeint.concepts.symplectic_system.valid_expressions"></a></span><a class="link" href="symplectic_system.html#boost_numeric_odeint.concepts.symplectic_system.valid_expressions">Valid
        expressions</a>
      </h5>
<div class="informaltable"><table class="table">
<colgroup>
<col>
<col>
<col>
<col>
</colgroup>
<thead><tr>
<th>
                <p>
                  Name
                </p>
              </th>
<th>
                <p>
                  Expression
                </p>
              </th>
<th>
                <p>
                  Type
                </p>
              </th>
<th>
                <p>
                  Semantics
                </p>
              </th>
</tr></thead>
<tbody>
<tr>
<td>
                <p>
                  Check for pair
                </p>
              </td>
<td>
                <p>
                  <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">is_pair</span><span class="special">&lt;</span>
                  <span class="identifier">System</span> <span class="special">&gt;::</span><span class="identifier">type</span></code>
                </p>
              </td>
<td>
                <p>
                  <code class="computeroutput"><span class="identifier">boost</span><span class="special">::</span><span class="identifier">mpl</span><span class="special">::</span><span class="identifier">true_</span></code>
                </p>
              </td>
<td>
                <p>
                  Check if System is a pair
                </p>
              </td>
</tr>
<tr>
<td>
                <p>
                  Calculate <span class="emphasis"><em>dq/dt = f(p)</em></span>
                </p>
              </td>
<td>
                <p>
                  <code class="computeroutput"><span class="identifier">sys</span><span class="special">.</span><span class="identifier">first</span><span class="special">(</span>
                  <span class="identifier">p</span> <span class="special">,</span>
                  <span class="identifier">dqdt</span> <span class="special">)</span></code>
                </p>
              </td>
<td>
                <p>
                  <code class="computeroutput"><span class="keyword">void</span></code>
                </p>
              </td>
<td>
                <p>
                  Calculates <span class="emphasis"><em>f(p)</em></span>, the result is stored into
                  <code class="computeroutput"><span class="identifier">dqdt</span></code>
                </p>
              </td>
</tr>
<tr>
<td>
                <p>
                  Calculate <span class="emphasis"><em>dp/dt = g(q)</em></span>
                </p>
              </td>
<td>
                <p>
                  <code class="computeroutput"><span class="identifier">sys</span><span class="special">.</span><span class="identifier">second</span><span class="special">(</span>
                  <span class="identifier">q</span> <span class="special">,</span>
                  <span class="identifier">dpdt</span> <span class="special">)</span></code>
                </p>
              </td>
<td>
                <p>
                  <code class="computeroutput"><span class="keyword">void</span></code>
                </p>
              </td>
<td>
                <p>
                  Calculates <span class="emphasis"><em>g(q)</em></span>, the result is stored into
                  <code class="computeroutput"><span class="identifier">dpdt</span></code>
                </p>
              </td>
</tr>
</tbody>
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<td align="right"><div class="copyright-footer">Copyright &#169; 2009-2015 Karsten Ahnert and Mario Mulansky<p>
        Distributed under the Boost Software License, Version 1.0. (See accompanying
        file LICENSE_1_0.txt or copy at <a href="http://www.boost.org/LICENSE_1_0.txt" target="_top">http://www.boost.org/LICENSE_1_0.txt</a>)
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